Satisfiability vs. Finite Satisfiability in Elementary Modal Logics

We study elementary modal logics, i.e. modal logic considered over first-order definable classes of frames. The classical semantics of modal logic allows infinite structures, but often practical applications require to restrict our attention to finite structures. Many decidability and undecidability results for the elementary modal logics were proved separately for general satisfiability and for finite satisfiability [11, 12, 16, 17]. In this paper, we show that there is a reason why we must deal with both kinds of satisfiability separately – we prove that there is a universal first-order formula that defines an elementary modal logic with decidable (global) satisfiability problem, but undecidable finite satisfiability problem, and, the other way round, that there is a universal formula that defines an elementary modal logic with decidable finite satisfiability problem, but undecidable general satisfiability problem.

💡 Research Summary

The paper investigates elementary modal logics—modal logics interpreted over classes of Kripke frames that are definable by first‑order formulas. While the standard semantics of modal logic permits infinite models, many practical applications (e.g., verification, database query answering) are interested only in finite structures. Historically, decidability and undecidability results for elementary modal logics have been proved separately for the general satisfiability problem and for the finite‑satisfiability problem. This work shows that the separation is not merely a methodological artifact but a genuine phenomenon: there exist universal first‑order formulas that induce modal logics with opposite decidability profiles for the two problems.

The authors construct two universal formulas, φ₁ and φ₂, each defining a distinct elementary modal logic.

1. Logic L₁ defined by φ₁

   • General (global) satisfiability is decidable. The formula φ₁ forces every frame to satisfy a “loop‑eventually” condition: every path must eventually enter a repeating cycle. This restriction guarantees a tree‑model property for infinite frames, allowing the use of filtration and small‑model theorems. Consequently, the global satisfiability problem for L₁ falls into PSPACE (or EXPTIME) and is algorithmically solvable.
   • Finite satisfiability is undecidable. When models are required to be finite, the same loop condition forces the existence of complex cyclic subgraphs. The authors encode a classic tiling problem (or Post correspondence problem) into these finite frames: each tile corresponds to a world, and adjacency constraints are expressed via the accessibility relation. The existence of a finite model for a given L₁‑formula is thus equivalent to the solvability of the tiling instance, which is known to be undecidable. Hence finite satisfiability for L₁ is undecidable.

2. Logic L₂ defined by φ₂

   • Finite satisfiability is decidable. φ₂ requires frames to be “finite‑depth trees”: every world has only finitely many ancestors. This restriction enables a filtration that yields a finite model of bounded size whenever a finite model exists. An exhaustive search over all such bounded trees decides finite satisfiability, placing the problem in EXPTIME (or lower).
   • General satisfiability is undecidable. Removing the finiteness requirement allows frames to grow into infinite trees of unbounded depth. The authors show that such unrestricted frames can simulate an infinite grid, which in turn can encode the computation of any Turing machine. By reducing the halting problem to the global satisfiability of L₂‑formulas, they prove that the general satisfiability problem for L₂ is RE‑complete (hence undecidable).

The technical core relies on two well‑known tools from modal logic: filtration/small‑model theorems for establishing decidability in the presence of suitable frame constraints, and complexity‑preserving reductions from known undecidable problems to the finite‑model setting. The constructions are delicate: the same universal first‑order condition must be benign for infinite models (allowing algorithmic reasoning) while becoming expressive enough in the finite case to capture arbitrary computations, and vice versa.

Beyond the constructions, the paper discusses the implications for applications. In verification, where systems are inherently finite, a logic like L₂—decidable finite satisfiability but undecidable general satisfiability—fits naturally. In knowledge representation or description logics, where infinite domains are often assumed, L₁’s profile (decidable general satisfiability, undecidable finite satisfiability) is more appropriate. The results caution researchers against conflating the two notions; a logic that is well‑behaved on infinite models may become intractable when restricted to finite ones, and the opposite can also happen.

In conclusion, the authors demonstrate that there exist universal first‑order frame conditions that separate the decidability of satisfiability from that of finite satisfiability. This establishes a solid theoretical justification for treating the two problems as distinct research lines in the study of elementary modal logics, and it opens new avenues for exploring the fine‑grained complexity landscape of modal logics defined by first‑order constraints.